DOCSLIB.ORG

The Properties of Partial Trace and Block Trace Operators of Partitioned Matrices∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Properties of Partial Trace and Block Trace Operators of Partitioned Matrices∗ Electronic Journal of Linear Algebra, ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 33, pp. 3-15, May 2018. THE PROPERTIES OF PARTIAL TRACE AND BLOCK TRACE OPERATORS OF PARTITIONED MATRICES∗ KATARZYNA FILIPIAKy , DANIEL KLEINz , AND ERIKA VOJTKOVA´ x Abstract. The aim of this paper is to give the properties of two linear operators defined on non-square partitioned matrix: the partial trace operator and the block trace operator. The conditions for symmetry, nonnegativity, and positive-definiteness are given, as well as the relations between partial trace and block trace operators with standard trace, vectorizing and the Kronecker product operators. Both partial trace as well as block trace operators can be widely used in statistics, for example in the estimation of unknown parameters under the multi-level multivariate models or in the theory of experiments for the determination of an optimal designs under the linear models. Key words. Partial trace operator, Block trace operator, Block matrix. AMS subject classifications. 15A15, 47B99. 1. Introduction. Throughout the paper we use \vec" operator which stacks the columns of the matrix one below another. The i-th column of an identity matrix Iu is denoted by ei;u, moreover recall that Pu 0 Pu Iu = i=1 ei;uei;u and vec Iu = i=1 ei;u ⊗ ei;u. In this paper we consider two linear operators from the mp × np dimensional space V of matrices, into the m × n space U of matrices, defined on specific partitions of A = (Aij) 2 V, where Aij are either p × p or m × n blocks of A. Definition 1.1. For an arbitrary matrix A = (Aij): mp × np the m × n partial trace operator, denoted by PTrp A, and the m × n block trace operator, denoted by BTrm;n A, can be defined in the following equivalent forms: (i) the matrix of the traces of p × p blocks of A, that is PTrp A = (tr Aij); i = 1; : : : ; m; j = 1; : : : ; n; and, respectively, the sum of all diagonal m × n blocks of A, that is p X BTrm;n A = Aii; i=1 Pp 0 (ii) PTrp A = i=1(Im ⊗ ei;p)A(In ⊗ ei;p), Pp 0 BTrm;n A = i=1(ei;p ⊗ Im)A(ei;p ⊗ In); ∗Received by the editors on December 22, 2017. Accepted for publication on February 26, 2018. Handling Editor: Heike Fassbender. yInstitute of Mathematics, Pozna´nUniversity of Technology, Poland (katarzyna.fi[email protected]). Supported by Statutory Activities No. 04/43/DSPB/0088. zInstitute of Mathematics, Faculty of Science, P. J. Saf´arikUniversity,ˇ Koˇsice,Slovakia ([email protected]). Supported by grants VEGA MSˇ SR 1/0344/14 and VEGA MSˇ SR 1/0073/15. xInstitute of Mathematics, Faculty of Science, P. J. Saf´arikUniversity,ˇ Koˇsice,Slovakia ([email protected]). Electronic Journal of Linear Algebra, ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 33, pp. 3-15, May 2018. K. Filipiak, D. Klein and E. Vojtkov´a 4 0 (iii) PTrp A = (Im ⊗ vec Ip)(A ⊗ Ip)(In ⊗ vec Ip), 0 BTrm;n A = (vec Ip ⊗ Im)(Ip ⊗ A)(vec Ip ⊗ In). Since both of the operators correspond to two different partitions of the mp×np matrix A, the subscripts used in the notation indicate the size of blocks of A due to the respective partition. For convenience, if n = m we denote BTrm;n A as BTrm A. The partial trace and block trace operators of square matrices have been studied in the physics and mathematics before, though not necessarily under these names and using different notations. De Pillis [3] studied mn × mn block matrices with blocks of order n. He showed that replacing every block of positive semi-definite matrix by its trace preserves positive semi-definiteness. Zhang [30] considered square block matrices with blocks replaced by their functions. Among others he proved the same result as de Pillis [3], denoting the block matrix with blocks replaced by their traces by T . Note, that the operation considered by de Pillis [3] as well as Zhang [30] correspond to the partial trace operator defined in Definition 1.1 (i) for particular case m = n, and the property of preserving positive semi-definiteness corresponds to the property given in Lemma 2.4 (iii) of this paper. For an arbitrary Kronecker product of two Hilbert spaces H1 and H2 of dimensions m and n, respectively, Bhatia [1] and Petz [21] defined two different matrix operations as linear maps from the space of linear operators on the Kronecker product H1 ⊗ H2 onto the space of linear operators on H1 and on H2. Both of these linear maps were called partial trace operators, and are widely used in physical sciences. Bhatia [1] denoted partial trace operators by A1 = trH1 A and by A2 = trH2 A, whilst Petz [21] used the notation Tr2 and Tr1, respectively. Both of these authors have shown several inequalities used in quantum mechanics. Note that the first and third definition of A1 of Bhatia [1, p. 126, 127] correspond respectively to the partial trace operator PTrp A given in Definition 1.1 (ii) and (i) for m = n. Moreover, the definitions of A2 of Bhatia [1, p. 126, 127] correspond to the partial trace operator PTrm A defined as in Definition 1.1 (ii) and (i) but with different partition of A (the blocks of A are m × m dimensional here). Petz [21] defined respective partial traces for a Kronecker product using the properties of standard trace operator. The definition of Tr2 corresponds to the property of partial trace operator given in Lemma 2.13, whilst the definition of Tr1 corresponds to the property of block trace operator given in Lemma 2.13, with m = n in both cases. It means that for A being a Kronecker product of two square matrices, H1 : m × m and H2 : p × p, we get BTrp(H1 ⊗ H2) = tr(H1)H2, thus, Tr1 of Petz [21] can be also expressed in terms of block trace operator given in Definition 1.1 with different partition, i.e., BTrp A. Recently, some inequalities related to partial trace operators defined by Bhatia [1] or Petz [21] were studied by Choi [2]. He considered partitioned matrices not necessarily having Kronecker product structure and denoted the operators respectively by tr2A and tr1A. Vit´oria[29] and Martins et al. [20] considered the block trace operator trb(Ab) of square block matrix Ab in context of differential equations. Recently, Jackson et al. [11] and Jackson et al. [12] used the block trace operator btr(A) of a square block matrix A in meta-analysis. Considering some statistical properties of physical model Nehorai and Paldi [19] defined the block trace operator btr[A] of a square matrix A. This definition corresponds in fact to the definition of partial trace given in Definition 1.1 (i). In statistical research Filipiak and Markiewicz [7,8] denoted the partial trace operator on an mp × mp matrix A by TrA and TrpA and used it to show optimality of some experimental designs. Moreover, Filipiak and Klein [5] proved some properties of partial trace operator to express the maximum likelihood estimators Electronic Journal of Linear Algebra, ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 33, pp. 3-15, May 2018. 5 The properties of partial trace and block trace operators of partitioned matrices of unknown parameters under the generalized growth curve model. Roy et al. [23] used the block trace operator, denoted by blktrA, for solving the problem of testing the mean vector for doubly multivariate observations. The same notation for block trace was used also by Roy et al. [24] for maximum likelihood estimation problem for doubly exchangeable covariance structure. In some statistical applications it is comfortable to use the partial trace of non-square matrices (see e.g. Section 3.1), which can be defined for every block matrix with square blocks. Since there is very strong relation between considered operators, and since also non-square matrices can be summed up, we have decided to define the block trace operator (as a generalization of the standard trace operator applied on the blocks of a matrix) also for non-square matrices. Therefore in this paper we extend the operators presented in the literature into the case of arbitrary matrix (not necessarily square). The aim of this paper is to unify the notation used for partial trace and block trace operators and to show the relation between them as well as some of their properties. We characterize some sequential properties of these two operators as well as some preserved (and not preserved) properties, such as symmetry, nonnegativity, positive semi-definiteness, M-matrix property, singularity, and commutativity. We also study the relations between considered operators and \vec" and Kronecker product operators, which are very useful especially in statistics. Finally, we show the conditions for which the partial trace and block trace operator of a product of matrices can be presented as the product of matrices and we present some properties in relation to Bhatia's [1] definition of partial trace operator. Both operators can be widely used for multi-level multivariate models in statistics, for example in the estimation of unknown parameters. They allow an elegant presentation of a complex formulas for estimators as well as an easier and faster computation of estimates. Possible applications of these operators in statistics are presented in Section3. 2. Properties of partial trace and block trace operators. Let Km;n be a commutation matrix defined as a matrix which transforms vec X into vec(X0) for any matrix X : m × n (for the details see e.g.
Load more

Recommended publications
  • Estimations of the Trace of Powers of Positive Self-Adjoint Operators by Extrapolation of the Moments∗
    Electronic Transactions on Numerical Analysis. ETNA Volume 39, pp. 144-155, 2012. Kent State University Copyright 2012, Kent State University. http://etna.math.kent.edu ISSN 1068-9613. ESTIMATIONS OF THE TRACE OF POWERS OF POSITIVE SELF-ADJOINT OPERATORS BY EXTRAPOLATION OF THE MOMENTS∗ CLAUDE BREZINSKI†, PARASKEVI FIKA‡, AND MARILENA MITROULI‡ Abstract. Let A be a positive self-adjoint linear operator on a real separable Hilbert space H. Our aim is to build estimates of the trace of Aq, for q ∈ R. These estimates are obtained by extrapolation of the moments of A. Applications of the matrix case are discussed, and numerical results are given. Key words. Trace, positive self-adjoint linear operator, symmetric matrix, matrix powers, matrix moments, extrapolation. AMS subject classifications. 65F15, 65F30, 65B05, 65C05, 65J10, 15A18, 15A45. 1. Introduction. Let A be a positive self-adjoint linear operator from H to H, where H is a real separable Hilbert space with inner product denoted by (·, ·). Our aim is to build estimates of the trace of Aq, for q ∈ R. These estimates are obtained by extrapolation of the integer moments (z, Anz) of A, for n ∈ N. A similar procedure was first introduced in [3] for estimating the Euclidean norm of the error when solving a system of linear equations, which corresponds to q = −2. The case q = −1, which leads to estimates of the trace of the inverse of a matrix, was studied in [4]; on this problem, see [10]. Let us mention that, when only positive powers of A are used, the Hilbert space H could be infinite dimensional, while, for negative powers of A, it is always assumed to be a finite dimensional one, and, obviously, A is also assumed to be invertible.
    [Show full text]
  • Block Kronecker Products and the Vecb Operator* CORE View
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Block Kronecker Products and the vecb Operator* Ruud H. Koning Department of Economics University of Groningen P.O. Box 800 9700 AV, Groningen, The Netherlands Heinz Neudecker+ Department of Actuarial Sciences and Econometrics University of Amsterdam Jodenbreestraat 23 1011 NH, Amsterdam, The Netherlands and Tom Wansbeek Department of Economics University of Groningen P.O. Box 800 9700 AV, Groningen, The Netherlands Submitted hv Richard Rrualdi ABSTRACT This paper is concerned with two generalizations of the Kronecker product and two related generalizations of the vet operator. It is demonstrated that they pairwise match two different kinds of matrix partition, viz. the balanced and unbalanced ones. Relevant properties are supplied and proved. A related concept, the so-called tilde transform of a balanced block matrix, is also studied. The results are illustrated with various statistical applications of the five concepts studied. *Comments of an anonymous referee are gratefully acknowledged. ‘This work was started while the second author was at the Indian Statistiral Institute, New Delhi. LINEAR ALGEBRA AND ITS APPLICATIONS 149:165-184 (1991) 165 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010 0024-3795/91/$3.50 166 R. H. KONING, H. NEUDECKER, AND T. WANSBEEK INTRODUCTION Almost twenty years ago Singh [7] and Tracy and Singh [9] introduced a generalization of the Kronecker product A@B. They used varying notation for this new product, viz. A @ B and A &3B. Recently, Hyland and Collins [l] studied the-same product under rather restrictive order conditions.
    [Show full text]
  • Genius Manual I
    Genius Manual i Genius Manual Genius Manual ii Copyright © 1997-2016 Jiríˇ (George) Lebl Copyright © 2004 Kai Willadsen Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License (GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link or in the file COPYING-DOCS distributed with this manual. This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the license. Many of the names used by companies to distinguish their products and services are claimed as trademarks. Where those names appear in any GNOME documentation, and the members of the GNOME Documentation Project are made aware of those trademarks, then the names are in capital letters or initial capital letters. DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE WITH THE FURTHER UNDERSTANDING THAT: 1. DOCUMENT IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES THAT THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE OF THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS WITH YOU.
    [Show full text]
  • 5 the Dirac Equation and Spinors
    5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.
    [Show full text]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • 18.700 JORDAN NORMAL FORM NOTES These Are Some Supplementary Notes on How to Find the Jordan Normal Form of a Small Matrix. Firs
    18.700 JORDAN NORMAL FORM NOTES These are some supplementary notes on how to find the Jordan normal form of a small matrix. First we recall some of the facts from lecture, next we give the general algorithm for finding the Jordan normal form of a linear operator, and then we will see how this works for small matrices. 1. Facts Throughout we will work over the field C of complex numbers, but if you like you may replace this with any other algebraically closed field. Suppose that V is a C-vector space of dimension n and suppose that T : V → V is a C-linear operator. Then the characteristic polynomial of T factors into a product of linear terms, and the irreducible factorization has the form m1 m2 mr cT (X) = (X − λ1) (X − λ2) ... (X − λr) , (1) for some distinct numbers λ1, . , λr ∈ C and with each mi an integer m1 ≥ 1 such that m1 + ··· + mr = n. Recall that for each eigenvalue λi, the eigenspace Eλi is the kernel of T − λiIV . We generalized this by defining for each integer k = 1, 2,... the vector subspace k k E(X−λi) = ker(T − λiIV ) . (2) It is clear that we have inclusions 2 e Eλi = EX−λi ⊂ E(X−λi) ⊂ · · · ⊂ E(X−λi) ⊂ .... (3) k k+1 Since dim(V ) = n, it cannot happen that each dim(E(X−λi) ) < dim(E(X−λi) ), for each e e +1 k = 1, . , n. Therefore there is some least integer ei ≤ n such that E(X−λi) i = E(X−λi) i .
    [Show full text]
  • Package 'Fastmatrix'
    Package ‘fastmatrix’ May 8, 2021 Type Package Title Fast Computation of some Matrices Useful in Statistics Version 0.3-819 Date 2021-05-07 Author Felipe Osorio [aut, cre] (<https://orcid.org/0000-0002-4675-5201>), Alonso Ogueda [aut] Maintainer Felipe Osorio <[email protected]> Description Small set of functions to fast computation of some matrices and operations useful in statistics and econometrics. Currently, there are functions for efficient computation of duplication, commutation and symmetrizer matrices with minimal storage requirements. Some commonly used matrix decompositions (LU and LDL), basic matrix operations (for instance, Hadamard, Kronecker products and the Sherman-Morrison formula) and iterative solvers for linear systems are also available. In addition, the package includes a number of common statistical procedures such as the sweep operator, weighted mean and covariance matrix using an online algorithm, linear regression (using Cholesky, QR, SVD, sweep operator and conjugate gradients methods), ridge regression (with optimal selection of the ridge parameter considering the GCV procedure), functions to compute the multivariate skewness, kurtosis, Mahalanobis distance (checking the positive defineteness) and the Wilson-Hilferty transformation of chi squared variables. Furthermore, the package provides interfaces to C code callable by another C code from other R packages. Depends R(>= 3.5.0) License GPL-3 URL https://faosorios.github.io/fastmatrix/ NeedsCompilation yes LazyLoad yes Repository CRAN Date/Publication 2021-05-08 08:10:06 UTC R topics documented: array.mult . .3 1 2 R topics documented: asSymmetric . .4 bracket.prod . .5 cg ..............................................6 comm.info . .7 comm.prod . .8 commutation . .9 cov.MSSD . 10 cov.weighted .
    [Show full text]
  • Trace Inequalities for Matrices
    Bull. Aust. Math. Soc. 87 (2013), 139–148 doi:10.1017/S0004972712000627 TRACE INEQUALITIES FOR MATRICES KHALID SHEBRAWI ˛ and HUSSIEN ALBADAWI (Received 23 February 2012; accepted 20 June 2012) Abstract Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then tr(AB)k ≤ minfkAkk tr Bk; kBkk tr Akg for any positive integer k. 2010 Mathematics subject classification: primary 15A18; secondary 15A42, 15A45. Keywords and phrases: trace inequalities, eigenvalues, singular values. 1. Introduction Let Mn(C) be the algebra of all n × n matrices over the complex number field. The singular values of A 2 Mn(C), denoted by s1(A); s2(A);:::; sn(A), are the eigenvalues ∗ 1=2 of the matrix jAj = (A A) arranged in such a way that s1(A) ≥ s2(A) ≥ · · · ≥ sn(A). 2 ∗ ∗ Note that si (A) = λi(A A) = λi(AA ), so for a positive semidefinite matrix A, we have si(A) = λi(A)(i = 1; 2;:::; n). The trace functional of A 2 Mn(C), denoted by tr A or tr(A), is defined to be the sum of the entries on the main diagonal of A and it is well known that the trace of a Pn matrix A is equal to the sum of its eigenvalues, that is, tr A = j=1 λ j(A). Two principal properties of the trace are that it is a linear functional and, for A; B 2 Mn(C), we have tr(AB) = tr(BA).
    [Show full text]
  • Dynamics of Correlation Structure in Stock Market
    Entropy 2014, 16, 455-470; doi:10.3390/e16010455 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Dynamics of Correlation Structure in Stock Market Maman Abdurachman Djauhari * and Siew Lee Gan Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor Bahru, Johor Darul Takzim, Malaysia; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +60-017-520-0795; Fax: +60-07-556-6162. Received: 24 September 2013; in revised form: 19 December 2013 / Accepted: 24 December 2013 / Published: 6 January 2014 Abstract: In this paper a correction factor for Jennrich’s statistic is introduced in order to be able not only to test the stability of correlation structure, but also to identify the time windows where the instability occurs. If Jennrich’s statistic is only to test the stability of correlation structure along predetermined non-overlapping time windows, the corrected statistic provides us with the history of correlation structure dynamics from time window to time window. A graphical representation will be provided to visualize that history. This information is necessary to make further analysis about, for example, the change of topological properties of minimal spanning tree. An example using NYSE data will illustrate its advantages. Keywords: Mahalanobis square distance; multivariate normal distribution; network topology; Pearson correlation coefficient; random matrix PACS Codes: 89.65.Gh; 89.75.Fb 1. Introduction Correlation structure among stocks in a given portfolio is a complex structure represented numerically in the form of a symmetric matrix where all diagonal elements are equal to 1 and the off-diagonals are the correlations of two different stocks.
    [Show full text]
  • Lecture 28: Eigenvalues Allowing Complex Eigenvalues Is Really a Blessing
    Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011 cos(t) sin(t) 2 2 For a rotation A = the characteristic polynomial is λ − 2 cos(α)+1 − sin(t) cos(t) which has the roots cos(α) ± i sin(α)= eiα. Lecture 28: Eigenvalues Allowing complex eigenvalues is really a blessing. The structure is very simple: Fundamental theorem of algebra: For a n × n matrix A, the characteristic We have seen that det(A) = 0 if and only if A is invertible. polynomial has exactly n roots. There are therefore exactly n eigenvalues of A if we count them with multiplicity. The polynomial fA(λ) = det(A − λIn) is called the characteristic polynomial of 1 n n−1 A. Proof One only has to show a polynomial p(z)= z + an−1z + ··· + a1z + a0 always has a root z0 We can then factor out p(z)=(z − z0)g(z) where g(z) is a polynomial of degree (n − 1) and The eigenvalues of A are the roots of the characteristic polynomial. use induction in n. Assume now that in contrary the polynomial p has no root. Cauchy’s integral theorem then tells dz 2πi Proof. If Av = λv,then v is in the kernel of A − λIn. Consequently, A − λIn is not invertible and = =0 . (1) z =r | | | zp(z) p(0) det(A − λIn)=0 . On the other hand, for all r, 2 1 dz 1 2π 1 For the matrix A = , the characteristic polynomial is | | ≤ 2πrmax|z|=r = . (2) z =r 4 −1 | | | zp(z) |zp(z)| min|z|=rp(z) The right hand side goes to 0 for r →∞ because 2 − λ 1 2 det(A − λI2) = det( )= λ − λ − 6 .
    [Show full text]
  • Approximating Spectral Sums of Large-Scale Matrices Using Stochastic Chebyshev Approximations∗
    Approximating Spectral Sums of Large-scale Matrices using Stochastic Chebyshev Approximations∗ Insu Han y Dmitry Malioutov z Haim Avron x Jinwoo Shin { March 10, 2017 Abstract Computation of the trace of a matrix function plays an important role in many scientific com- puting applications, including applications in machine learning, computational physics (e.g., lat- tice quantum chromodynamics), network analysis and computational biology (e.g., protein fold- ing), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estima- tors (Hutchinson’s method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algo- rithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten p-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions. 1 Introduction Given a symmetric matrix A 2 Rd×d and function f : R ! R, we study how to efficiently compute d X Σf (A) = tr(f(A)) = f(λi); (1) i=1 arXiv:1606.00942v2 [cs.DS] 9 Mar 2017 where λ1; : : : ; λd are eigenvalues of A.
    [Show full text]
  • Math 217: True False Practice Professor Karen Smith 1. a Square Matrix Is Invertible If and Only If Zero Is Not an Eigenvalue. Solution Note: True
    (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 217: True False Practice Professor Karen Smith 1. A square matrix is invertible if and only if zero is not an eigenvalue. Solution note: True. Zero is an eigenvalue means that there is a non-zero element in the kernel. For a square matrix, being invertible is the same as having kernel zero. 2. If A and B are 2 × 2 matrices, both with eigenvalue 5, then AB also has eigenvalue 5. Solution note: False. This is silly. Let A = B = 5I2. Then the eigenvalues of AB are 25. 3. If A and B are 2 × 2 matrices, both with eigenvalue 5, then A + B also has eigenvalue 5. Solution note: False. This is silly. Let A = B = 5I2. Then the eigenvalues of A + B are 10. 4. A square matrix has determinant zero if and only if zero is an eigenvalue. Solution note: True. Both conditions are the same as the kernel being non-zero. 5. If B is the B-matrix of some linear transformation V !T V . Then for all ~v 2 V , we have B[~v]B = [T (~v)]B. Solution note: True. This is the definition of B-matrix. 21 2 33 T 6. Suppose 40 2 05 is the matrix of a transformation V ! V with respect to some basis 0 0 1 B = (f1; f2; f3). Then f1 is an eigenvector. Solution note: True. It has eigenvalue 1. The first column of the B-matrix is telling us that T (f1) = f1.
    [Show full text]